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Three friends, Jessa, Tyree, and Ben, are collecting canned food for a culinary skills class. Their canned food collection goal is represented by the expression [tex]\(10x^2 - 4xy + 12\)[/tex]. The friends have already collected the following number of cans:

- Jessa: [tex]\(7xy + 3\)[/tex]
- Tyree: [tex]\(3x^2 - 4\)[/tex]
- Ben: [tex]\(5x^2\)[/tex]

Part A: Write an expression to represent the amount of canned food collected so far by the three friends. Show all your work. (5 points)

Part B: Write an expression that represents the number of cans the friends still need to collect to meet their goal. Show all your work. (5 points)



Answer :

GinnyAnswer
Sure, let's solve this problem step by step.

### Part A: Expression for the Amount of Canned Food Collected So Far

Each of the friends has collected a certain number of cans, given by the following expressions:

- Jessa collected: [tex]\( 7xy + 3 \)[/tex] cans
- Tyree collected: [tex]\( 3x^2 - 4 \)[/tex] cans
- Ben collected: [tex]\( 5x^2 \)[/tex] cans

To find the total amount of canned food collected by all three friends so far, we need to add these expressions together:

[tex]\[ (7xy + 3) + (3x^2 - 4) + (5x^2) \][/tex]

Now we combine the like terms:

1. The [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 5x^2 = 8x^2\)[/tex]
2. The [tex]\(xy\)[/tex] term: [tex]\(7xy\)[/tex] (as there is no other [tex]\(xy\)[/tex] term to combine with)
3. The constants: [tex]\(3 - 4 = -1\)[/tex]

So, the total amount of canned food collected so far is:

[tex]\[ 8x^2 + 7xy - 1 \][/tex]

### Part B: Expression for the Number of Cans Still Needed to Meet Their Goal

The goal for the canned food collection is represented by the expression [tex]\( 10x^2 - 4xy + 12 \)[/tex]. We already found that the total amount collected so far is [tex]\( 8x^2 + 7xy - 1 \)[/tex].

To find out how many more cans they need to collect, we need to subtract the total collected from the goal:

[tex]\[ (10x^2 - 4xy + 12) - (8x^2 + 7xy - 1) \][/tex]

Distribute the subtraction:

[tex]\[ 10x^2 - 4xy + 12 - 8x^2 - 7xy + 1 \][/tex]

Now combine the like terms:

1. The [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 - 8x^2 = 2x^2\)[/tex]
2. The [tex]\(xy\)[/tex] terms: [tex]\(-4xy - 7xy = -11xy\)[/tex]
3. The constants: [tex]\(12 + 1 = 13\)[/tex]

So, the number of cans needed to meet the goal is:

[tex]\[ 2x^2 - 11xy + 13 \][/tex]

### Summary

Part A:

The expression representing the amount of canned food collected so far is:

[tex]\[ 8x^2 + 7xy - 1 \][/tex]

Part B:

The expression representing the number of cans still needed to meet their goal is:

[tex]\[ 2x^2 - 11xy + 13 \][/tex]

These expressions give the total number of canned foods collected so far and the additional number of cans needed to meet the goal, respectively.

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